Do the following using R. You must also turn in a copy of your R code. (1) Plot the Binomial(n = 47, θ = 0.8) pmf. Make sure the plot is properly labeled.

n <- 47
theta <- 0.8
X <- 1:n
fX <- dbinom(X,n,theta)
plot(X, fX, type = 'h', lwd = 3, main = 'X~Binom(47,0.8)', xlab = 'X', ylab = 'f(x) (Density)')

2. What is the expected value of a Binomial(n=47, θ = 0.8) random variable?

n*theta

## [1] 37.6

3. What is the standard deviation of a Binomial(n = 47, θ = 0.8) random variable?

sqrt(n*theta*(1-theta))

## [1] 2.742262

4. Following are data from a clinical trial involving chemotherapy for testicular cancer. It is considered a success if a patient survived five years past the clinical trial. 3 subjects survived for five years past the trial, while 8 did not. What is the value of the (binomial) likelihood that 3 successes occur in 11 trials if θ (the probability of survival) is 0.5?

dbinom(3,11,.5)

## [1] 0.08056641

5. Refer to question 4. What is the value of the likelihood if θ is 0.31?

dbinom(3,11,.31)

## [1] 0.2525584

6. Refer to question 4. What is the value of the likelihood if θ is 0.27?

dbinom(3,11,.27)

## [1] 0.2619136

7. Refer to question 4. Justify why you cannot find a value for θ that makes the likelihood larger than when θ = 3/11. The MLE of a binomial is p_hat = x/n
   

   Insert Proof

8. Plot a beta(3, 5) pdf and a beta(5, 11) pdf on the same graph; the beta(3,5) should be in black and the beta(5,11) in gray. Also, be sure to properly format this graphic (x- and y-axes, overall title, and a legend).

X <- seq(0,1,length.out=1001)
plot(X,dbeta(X,shape1 = 5, shape2 = 11), 
     col = 'black', lwd = 2, type = 'l',
     main = 'Beta(3,5) & Beta(5,11)',
     xlab = 'X',
     ylab = 'f(x)')
lines(X,dbeta(X, shape1 = 3, shape2 = 5),
      col = 'gray', lwd = 2)
legend(.75, 3, legend=c("Beta(5,11)", "Beta(3,5)"),
       col=c("black", "gray"), lty = 1, lwd = 2, cex = 1)

9. What is the height of the curve of a beta (1, 8) distribution at x = .24? That is, what is the pdf evaluated at x=.24?

dbeta(.24, shape1 = 1, shape2 = 8)

## [1] 1.171616

10. What is the probability a beta (1, 8) random variable is less than 0.13?

pbeta(.13, shape1 = 1, shape2 = 8)

## [1] 0.6717883

11. What is the probability a beta (3, 9) random variable is greater than .4?

1 - pbeta(.4, shape1 = 3, shape2 = 9)

## [1] 0.1189168

12. What is the probability a beta (18,4.4) random variable is between 0.6 and 0.7?

pbeta(.7,shape1 = 18, shape2 = 4.4) - pbeta(.6,shape1 = 18, shape2 = 4.4)

## [1] 0.09770375

13. At what value of x is the probability that a beta (4, 7) random varable is less than x equal to .71? That is, for what x is Pr(beta(4, 7) random variable < x) = .71?

qbeta(.71, shape1 = 4, shape2 = 7)

## [1] 0.438946

14. At what value of x is the probability that a beta (12.2, 25.7) random variable is less than x equal to .2? That is, for what x is P r(beta(12.2, 25.7) random variable < x) = .2?

qbeta(.2, shape1 = 12.2, shape2 = 25.7)

## [1] 0.2572049

15. What is the expected value of a beta (3.1,4.8) random variable?

a <- 3.1
b <- 4.8
a / (a+b)

## [1] 0.3924051

16. What is the variance of a beta (3, 5) random variable?

a <- 3
b <- 5
(a*b) / (((a+b)**2) * ((a+b+1)))

## [1] 0.02604167

17. What is the mode of a beta(2.8, 2.1) random variable?

a <- 2.8
b <- 2.1
(a−1) / (a+b−2)

## [1] 0.6206897

18. Show that the mode of a beta(a,b) random variable is \((a−1) / (a+b−2)\) when a > 1 and a + b > 2.
    

    Insert Proof

19. Refer again to question 4. In general terms, what is the posterior distribution for θ given the results of the clinical trial described in question 4 and assuming a beta(a,b) prior distribution for θ.

20. Refer again to question 4. For this problem and assuming a beta(a,b) prior, what is \[\int_0^1 f(y|θ)π(θ)~dθ \]

21. Refer again to question 4. Create one figure with two plots (i.e., use par(mfrow=c(1,2))). Each plot will contain two curves, a red one representing the prior distribution of θ and a blue representing the posterior distribution of θ. For the left plot, use a beta(1,1) prior distribution. Comment on what this prior distribution implies regarding prior beliefs of the analyst and the impact it has on the posterior. For the right plot, use a beta(25,1) prior distribution. Comment on what this prior distribution implies regarding prior beliefs of the analyst and the impact it has on the posterior.